EMAT 6680

Lissajous curves are the family of curves described
by the parametric equations

x(t) = a sin(nt +c)

y(t) = bsin(t)

The curves are named after Jules-Antonie Lissajous
(1822-1880) who discovered the curves (in 1857) while doing experiments involving
sound vibrations.They are also sometimes referred
to as Bowditch curves after the American mathematician Nathaniel Bowditch(1773-1838).He is said to have discovered the curves while studying
pendulums in 1815.These curves have applications in
physics, astronomy, mathematics and other sciences.

Investigation:Consider
the Lissajous curves of the form x(t) = 4sin((p/q)t) and y(t) = 3sin(t).
Graph these curves for various values of p and q where0 < p/q < 1.Finally,
compare the graphs of these curves with the graphs formed by the equations
x(t) = sin(pt) and y(t) = sin(qt).

Let’s begin our investigation by looking at some curves.We
will start our investigation by assigning p=1 for all of the curves and vary
only the q value.

Graph of curve when p = 1 and q =2

This graph is a closed figure with a “bow-tie”
appearance that reaches 2 peaks.

Graph of curve when p = 1 and q = 3

This graph is not closed and appears to have a
cubic shape.

Graph of curve when p =1 and q = 4

This graph is a closed figure and has 4 peaks.

Graph of curve when p = 1 and q = 5

This graph is not closed and appears to have the
shape of the graph of a degree five polynomial.

Graph of curve when p = 1 and q =6

This graph is a closed figure and reaches 6 peaks.

As we continue graphing equations of this form, we
see that a clear pattern develops.When q is an even
number the curve is closed and it reaches q peaks.For
example, when p =1 and q=4 the graph had 4 peaks.When
q is an odd number the curve is not closed and, if the endpoints are counted,
it reaches (q+1)/2 peaks. For example, when p = 1 and q = 5, the graph reached
(5 + 1)/2 or 3 peaks.

Next, let’s investigate what happens when both
p and q vary. First let’s look at the case where one number is odd while
the other is even.Remember that we are restricting
our investigation to values of p/q that are less than one.

Graph of curve when p = 2 and q = 3

This graph is a closed figure reaching three peaks
on top (and bottom) and with two “bumps” on the right and left sides.

Graph of curve when p
= 3 and q = 4

This graph is a closed figure reaching four peaks on top and with three“bumps”
on the side.

Graph of curve when p = 5 and q = 6

This graph is a closed figure reaching six peaks
on top and with five “bumps” on the side.

As we continue graphing equations of this form, we
see that a clear pattern develops.When q is an even
number and p is odd, or when q is odd and p is even, then the curve is closed.We can also see that the graph reaches q peaks on top
and bottom while it has p “bumps” on the side.For
example, when p =2 and q=3 the graph had 3 peaks and 2 bumps.

Now let’s investigate graphs formed when both p
and q are odd.

Graph of curve with p = 3 and q = 5

The graph is an open figure with three peaks and
two “bumps” on the side. (Remember to count the endpoints)

Graph of curve with p=5 and q = 7

The graph is an open figure with 4 peaks and three
“bumps” on the side.

It appears that when p and q are odd numbers the
curve is not closed.It also appears that these curves
reach (q+1)/2 peaks and have (p+1)/2 bumps, if the endpoints are counted.

Let’s take a closer
look at the peaks and bumps of the curves formed when p is odd and q is even
or vice-versa.To do this we will follow the equation
when p = 3 and q = 4 for changing values of t.We
will also graph the lines y = 3 and x = 4 for reference.

Graph of curve with p = 3, q = 4 and t =5

Notice that, as
the path of the graph is traced, it touches both lines in exactly one place.

Graph of curve with p = 3, q = 4 and t =10

Notice that the
graph has intersected the line y=3 a second time.

Graph of curve with p = 3, q = 4 and t =15

Graph of curve with p = 3, q = 4 and t =20

Graph of curve with p = 3, q = 4 and t =25

Graph of curve with p = 3, q = 4 and t =30

By adding the lines y = 3 and x = 4 to our graph
we can see that the peaks and bumps are actually points of tangency to these
two lines.As this graph was traced it touched the
y = 3 line four times while it touched the x = 4 line 3 times.Also, notice that the graph is symmetric to both the x
and y axis so that y = -3 and x =-4 are also tangent lines to the curve.

Let’s take a closer look at the peaks and bumps
of the curves formed when both p and q are odd.To
do this we will follow the equation when p = 3 and q = 5 for changing values
of t.We will also graph the lines y = 3 and x = 4
for reference.

Graph of curve with p = 3, q = 5 and t
= 5

Notice that, as the path of the graph is traced,
it touches both lines in exactly one place.

Graph of curve with p = 3, q = 5 and t = 10

Graph of curve with p = 3, q = 5 and t = 15

Note:This graph appears
to be the same as the graph for t = 10 but on closer examination it can be
seen that the graph is actually retracing itself.

Graph of curve with p = 3, q = 5 and t = 20

Graph of curve with p
= 3, q = 5 and t = 25

By adding the lines y = 3 and x = 4 to our graph
we can see that the peaks and bumps are actually points of tangency to these
two lines.As this graph was traced (and retraced)
it touched the y = 3 line five times while it touched the x = 4 line 3 times.Also, notice that although the graph is not symmetric
to the x and y axis, the lines y = -3 and x =-4 are also tangent lines to
the curve.

So what happens as the value of p/q approaches one?Let’s
investigate.

Graph of the curve with p = 11 and q = 12

Graph of the curve with p =20 and q = 21

Graph of the curve with p = 39 and q =40.

It appears that as the value of p/q approaches one,
the graph of the curve becomes a rectangle with dimension 4 units by 3 units,
the coefficients of the equation.

Finally,lets compare our
original graphs to the graphs formed by the equations

x = 4 sin((p)t) and y = 3sin((q)t).

Graph of curve when p = 1 and q = 2.

This curve resembles the original curve formed
when p = 1 and q = 2.

Graph of curve when p = 2 and q = 3.

This curve resembles the original curve formed
when p = 2 and q =3.

Graph of curve when p =3 and q =5

Once again we have a match!

It appears that the graphs formed by the parametric equation x = 4 sin((p)t)
and y = 3sin((q)t) will yield the same results as those constructed from
x(t) = 4sin((p/q)t) and y(t) = 3sin(t).